Theorem upfval 49666

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)

Assertion

Ref	Expression
upfval	⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐶,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐷,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐸,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐻,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐽,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝑂,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦

Proof of Theorem upfval

Dummy variables 𝑏 𝑐 𝑑 𝑒 ℎ 𝑗 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6842	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
2		fveq2 6827	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Base‘𝑑) = (Base‘𝐷))
3	2	adantr 481	. . . . 5 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
4		upfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
5	3, 4	eqtr4di 2792	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
6		fvexd 6842	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) ∈ V)
7		simplr 774	. . . . . . 7 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
8	7	fveq2d 6831	. . . . . 6 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
9		upfval.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
10	8, 9	eqtr4di 2792	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
11		fvexd 6842	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) ∈ V)
12		simplll 780	. . . . . . . 8 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → 𝑑 = 𝐷)
13	12	fveq2d 6831	. . . . . . 7 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		upfval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
15	13, 14	eqtr4di 2792	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = 𝐻)
16		fvexd 6842	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) ∈ V)
17		simp-4r 789	. . . . . . . . 9 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → 𝑒 = 𝐸)
18	17	fveq2d 6831	. . . . . . . 8 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = (Hom ‘𝐸))
19		upfval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐸)
20	18, 19	eqtr4di 2792	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = 𝐽)
21		fvexd 6842	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) ∈ V)
22		simp-5r 791	. . . . . . . . . 10 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → 𝑒 = 𝐸)
23	22	fveq2d 6831	. . . . . . . . 9 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = (comp‘𝐸))
24		upfval.o	. . . . . . . . 9 ⊢ 𝑂 = (comp‘𝐸)
25	23, 24	eqtr4di 2792	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = 𝑂)
26		simp-6l 792	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑑 = 𝐷)
27		simp-6r 793	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑒 = 𝐸)
28	26, 27	oveq12d 7374	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
29		simp-4r 789	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑐 = 𝐶)
30		simp-5r 791	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑏 = 𝐵)
31	30	eleq2d 2825	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
32		simplr 774	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑗 = 𝐽)
33	32	oveqd 7373	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝑓)‘𝑥)))
34	33	eleq2d 2825	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))))
35	31, 34	anbi12d 638	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)))))
36	32	oveqd 7373	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝑓)‘𝑦)))
37		simplr 774	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
38	37	oveqdr 7384	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
39		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑜 = 𝑂)
40	39	oveqd 7373	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)))
41	40	oveqd 7373	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))
42	41	eqeq2d 2750	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
43	38, 42	reueqbidv 3380	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
44	36, 43	raleqbidv 3313	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
45	30, 44	raleqbidv 3313	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
46	35, 45	anbi12d 638	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))))
47	46	opabbidv 5138	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
48	28, 29, 47	mpoeq123dv 7431	. . . . . . . 8 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
49	21, 25, 48	csbied2 3868	. . . . . . 7 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
50	16, 20, 49	csbied2 3868	. . . . . 6 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → ⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
51	11, 15, 50	csbied2 3868	. . . . 5 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
52	6, 10, 51	csbied2 3868	. . . 4 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
53	1, 5, 52	csbied2 3868	. . 3 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
54		df-up 49664	. . 3 ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
55		ovex 7389	. . . 4 ⊢ (𝐷 Func 𝐸) ∈ V
56	9	fvexi 6841	. . . 4 ⊢ 𝐶 ∈ V
57	55, 56	mpoex 8021	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) ∈ V
58	53, 54, 57	ovmpoa 7511	. 2 ⊢ ((𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
59		reldmup 49665	. . . 4 ⊢ Rel dom UP
60	59	ovprc 7394	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = ∅)
61		reldmfunc 49565	. . . . . 6 ⊢ Rel dom Func
62	61	ovprc 7394	. . . . 5 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 Func 𝐸) = ∅)
63	62	orcd 879	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → ((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅))
64		0mpo0 7439	. . . 4 ⊢ (((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
65	63, 64	syl 17	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
66	60, 65	eqtr4d 2777	. 2 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
67	58, 66	pm2.61i 183	1 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃!wreu 3342  Vcvv 3431  ⦋csb 3831  ∅c0 4261  ⟨cop 4561  {copab 5134  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17170  Hom chom 17222  compcco 17223   Func cfunc 17812   UP cup 49663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-func 17816  df-up 49664

This theorem is referenced by:  upfval2  49667  uppropd  49671  reldmup2  49672  relup  49673  uprcl  49674